Let M be a domain enclosed between two principal orbits on a cohomogeneity one manifold M1. Suppose that T and R are symmetric invariant (0, 2)-tensor fields on M and ∂M, respectively. The paper studies the prescribed Ricci curvature equation Ric(G)=T for a Riemannian metric G on M subject to the boundary condition G∂M=R (the notation G∂M here stands for the metric induced by G on ∂M). Imposing a standard assumption on M1, we describe a set of requirements on T and R that guarantee global and local solvability.